The left internal Hom of pointed sets satisfies the following universal property:
\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright )\cong \mathsf{Sets}_{*}\webleft (X,\webleft [Y,Z\webright ]^{\lhd }_{\mathsf{Sets}_{*}}\webright ) \]
That is to say, the following data are in bijection:
- Pointed maps $f\colon X\lhd Y\to Z$.
- Pointed maps $f\colon X\to \webleft [Y,Z\webright ]^{\lhd }_{\mathsf{Sets}_{*}}$.
